In this paper, we consider the problem of evaluating the rigidity of a planar network, while satisfying common objectives of real-world systems: decentralization, asynchronicity, and parallelization. Motivating our investigation are the implications that rigidity has in fundamental multi-robot problems, e.g. guaranteed formation stability and relative localizability. We propose the decentralization of the pebble game algorithm of Jacobs et. al., an O(n^2) method that determines the generic rigidity of a planar network. Our decentralization is based on asynchronous inter-agent message-passing and a distributed memory architecture, coupled with distributed auctions for electing network leaders to arbitrate evaluation. Further, we provide a parallelization of our methods that takes inspiration from gossip algorithms to yield significantly reduced execution time and messaging burden. A thorough analysis of the correctness, finite termination, and complexity of our propositions is given, along with a simulated application in decentralized rigidity control. Finally, we provide Monte Carlo analysis of our algorithms in a Contiki networking environment, illustrating the real-world applicability of our methods, and yielding a bridge between rigidity theory and realistic interacting systems.

This video demonstrates the application of our decentralized rigidity evaluation to the control of a mobile multi-robot team. Specifically, we assume that each of n = 7 mobile robots employs the constrained interaction framework (see Williams:TRO:2013) to allow for spatial topology control, specifically such that rigidity can be dynamically regulated. The robots are initialized in a highly connected configuration and apply a common dispersive control law to force link deletions, retaining only those links computed by application of the decentralized pebble game (red). When a minimally rigid network is reached, the robots enter into a path following phase that intersects the formation with a stationary obstacle. The obstacle forces the formation to split for passage, resulting in a non-rigid configuration. With only a rigidity maintenance control, the possibility of obstacle navigation is uncertain, and the non-rigidity after obstacle avoidance is undetectable. However, the proposed decentralized rigidity test allows the network to detect the non-rigidity and initiate a recovery maneuver, in this case an aggregative control that forces link additions. After rigidity is recovered, a dispersive control is reapplied to reach a minimally rigid network and continue path following.

This video demonstrates the application of our decentralized rigidity evaluation to the control of a mobile multi-robot team. Specifically, we assume a system of n = 7 mobile robots, each applying a standard dispersive control to force link deletion in the network. When an equilibrium position is reached, the network graph (rigid by control) is used to define a target formation, and the objective is switched to a formation keeping controller. As rigidity is a sufficient condition for formation stability, such a composite objective can be seen as growing rigid formations, in a dynamic and distributed way (contrasting with centralized and static methods in the literature), integrating adaptability with standard formation control techniques.